# If cotA+cotB+cotC=3^1/2 the prove that the triangle is isosceles triangle.The question is related to properties of triangle lesson....]

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This question is wrong. It should be an equilateral triangle. I will prove this for an equilateral triangle.

In an equilateral triangle `A=B=C` and `A+B+C = pi`

Therefore, `A=B=C= pi/3`

Therefore,

`cot(A)+cot(B)+cot(C) = cot(pi/3)+cot(pi/3)+cot(pi/3)`

`= 1/sqrt(3)+1/sqrt(3)+1/sqrt(3)`

`= 3/sqrt(3)`

`= sqrt(3)`

**Therefore if ABC is an equilateral triangle,**

`cot(A)+cot(B)+cot(C) =sqrt(3)`

And vice versa.

this is for an equilateral triangle.

if its an equilateral triangle then A=B=C and also A+B+C=pi

3A (or 3B or 3C)=pi

=>A (or B or C)=pi/3

cotA +cotB +cotC=cot(pi/3)+cot(pi/3)+cot(pi/3)

=3cot(pi/3)

=3cot60

=3/(3)^1/2

=3^1/2

hence it is an equilateral triangle

HENCE PROVED